In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible[1] or, equivalently, the geometric median of the three vertices.
It is so named because this problem was first raised by Fermat in a private letter to Evangelista Torricelli, who solved it.
The Fermat point gives a solution to the geometric median and Steiner tree problems for three points.
Construction
The Fermat point of a triangle with largest angle at most 120° is simply its first isogonic center or X(13)[citation needed], which is constructed as follows:
- Construct an equilateral triangle on each of two arbitrarily chosen sides of the given triangle.
- Draw a line from each new vertex to the opposite vertex of the original triangle.
- The two lines intersect at the Fermat point.
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